BEA BEAMS ND WH WHERE AND TO FIND ND THEM TO The Th e Gumb - - PowerPoint PPT Presentation

BEA BEAMS

Th The e Gumb umbel el-To Top-𝒍 Tri Trick for r Sampl mpling Se Sequ quences With thou

• ut

t Repla lacement

Wo Wouter er K Kool, He Herke van Ho Hoof, Max Welling

• “Yo

“You wi will ll ne

neve ver hav

have du duplic icate te samples es aga gain in!” !”

AND ND WH

WHERE

TO TO FIND

ND THEM

STO STOCH CHASTIC STIC

HONO NORABLE MENT NTION

ICM CML

2019 19

• “T

“Thi his is ho how w you

ra randomize a

a be beam am sear arch! h!”

TL TL;DR DR

Stoch Stochasti tic B c Bea eam Sea Search ch fi finds a a s set of et of un unique ue sampl ples es (w (without replacement) ) fr from a a s sequen equence m ce model el.

(log-)probability

Exa Example

Binarese language model

Vocabulary: {Abra, Cadabra}

What if we want a sam sampl ple from

• ur model?

𝑄(𝐷) 𝑄(𝐵|𝐷)

𝑄(𝐷|𝐷𝐵)

𝑄 𝐷 𝑄 𝐵 𝐷 𝑄 𝐷 𝐷𝐵 = 𝑄 𝐷𝐵𝐷

Th The G Gumbe bel-Max Max Tr Trick ck 𝐻* ∼ Gumbel(0) Gumbel noise 𝜚* = log 𝑞* log-probability 𝐻78 ∼ Gumbel 𝜚*

perturbed log-probability

= +

(Gumbel, 1945; Maddison et al., 2014)

“Prof. Gumbeldore”

Th The G Gumbe bel-Max Max Tr Trick ck

∼ Gumbel log :

*

exp 𝜚* argmax

*

𝐻78

max and argmax are independent

max

*

𝐻78 ∼ Categorical 𝑞* 𝐽 = 𝑄 𝐽 = 𝑗 = 𝑞*

(Gumbel, 1945; Maddison et al., 2014)

“Prof. Gumbeldore”

(log-)probability

Exa Example

Binarese language model

Vocabulary: {Abra, Cadabra}

This will be

• ur sample!

What if we want a sam sampl ple from

• ur model?

What happens if, instead of 1 (one), we take the 𝑙 largest elements (top 𝑙)?

𝐽F, … , 𝐽I = arg top 𝑙

*

𝐻78 𝑙 = 3

Th The ‘ ‘Gumbe bel-To Top-𝑙’ ’ Trick This is equivalent to repeated sampling without replacement!

𝑄 𝐽F = 𝑗F, … , 𝐽I = 𝑗I = 𝑞*K ⋅

M8N FOM8K ⋅ … ⋅ M8P FO∑ℓSK

PTK M8ℓ

= ∏VWF

I M8X FO∑ℓSK

XTK M8ℓ

(Vieira, 2014) Also known as Plackett-Luce

𝐽F, … , 𝐽I = arg top 𝑙

*

𝐻78

(log-)probability

Exa Example

Binarese language model

Vocabulary: {Abra, Cadabra}

This will be our set of samples!

We can get a set of unique samples from our model!

PR PROBLEM

… but we don’t have to! In general, constructing the full tree is not possible…

𝐻7Y = max

𝑻

Pe Pert rturb rbed log-pr probability ty

• f
• f partial seq

equen ence e (“C”)

Look at maximum of perturbed log-probabilities in subtree

𝜚[ = log-probability of “C” ∼ Gumbel log :

exp 𝜚*

Noise 𝐻[ ∼ Gumbel(0) is inferred

We can sample 𝐻7Y ∼ Gumbel 𝜚[ di directly ly

Start from root, sample 𝐻7Y ∼ Gumbel(𝜚[) Sample children 𝐻7Y]conditionally on max

[]∈^_`abcde([) 𝐻7Y] = 𝐻7Y

1. sample 𝐻7Y] independently, compute Z = max

2. ‘shift’ Gumbels in (negative) exponential space:

g 𝐻7Y]

To Top-dow down sam sampl pling

… the result is eq equiv ivalen ent to sampling Gjk for leaves directly!

(Maddison et al., 2014)

Th The K Key I Insight

We only need to expand the top 𝑙 nodes at each level in the tree

Threshold Each top 𝑙 node generates (at least)

• ne leaf (maximum) above threshold

Other nodes only generate leafs below threshold No need to expand At least 𝑙 leafs will be above threshold

We only need to expand the top 𝑙 nodes at each level in the tree

This is a beam search

Stoc Stochasti stic B Beam Se Search

Top 𝑙 according to perturbed log-probability = Sampling (without replacement)

Gumbel-Top-𝑙 trick

• A beam search that samples the nodes to expand
• But… samples children conditionally on parent
• The result is a sample without replacement

from the full sequence model

• Is a generalization of ancestral sampling (𝑙 = 1)

Important!

Stoc Stochasti stic B Beam Se Search

Ex Experim iments ts

• Generate 𝑙 translations
• Plot BLEU against diversity
• Vary softmax temperature
• Compare:
• Beam Search
• Stochastic Beam Search
• Sampling
• Diverse Beam Search

(Vijayakumar et al., 2018)

Tr Tran anslat ation D Diversity

BL BLEU S Scor

• re E

Est stimat ation

• n
• Estimate expected sentence-

level BLEU

• Plot mean and 95% interval
• vs. num samples
• Compare:
• Monte Carlo Sampling
• Stochastic Beam Search with

(normalized) Importance Weighted estimator

• Beam Search with

deterministic estimate

BEA BEAMS

AND ND WH

WHERE

TO TO FIND

ND THE

PO POST STER?

STO STOCH CHASTIC STIC

#4 #41

Wo Wouter er K Kool, He Herke van Ho Hoof, Max Welling